Learning From Reviews: Endogenous Histories and Information Bounds
A comment on Acemoglu, Makhdoumi, Malekian, and Ozdaglar (2022)
Abstract
Acemoglu, Makhdoumi, Malekian, and Ozdaglar (2022) characterize the learning rate from a terminal review summary by a Kullback–Leibler divergence that evaluates the true state at its limiting belief and the alternative state at the opposite limiting belief. A terminal summary is a function of the review history generated under the same summary-based policy. Data processing yields a finite-horizon information bound, and any constant almost-sure learning exponent is bounded by the information in that history. The history's asymptotic rate evaluates both states at the same limiting belief. Uniform primitives satisfying the paper's assumptions make the stated formulas equal 0.2176 and 0.2662, while the corresponding information bounds are 0.0288 and 0.1181. The example has positive selection, so it also contradicts the strict speed comparison in Proposition 2(2). Terminal-event likelihoods admit an exact finite-horizon relative-entropy representation. A model-specific large-deviation contraction of this representation supplies the corresponding asymptotic rate problem.
Technical point
A terminal-summary learning rate must compare the two state laws along the same endogenous history and belief path; the cross-boundary Kullback–Leibler divergences in Theorem 4 can exceed the information available in that history.
Scope
Claims affected
The two summary-statistic rate formulas in Theorem 4, the strict summary-over-full-history comparison in Proposition 2(2), and extensions that import the same cross-boundary rate.
What remains intact
Theorem 3 continues to deliver complete learning under strict separation, and Theorem 2 continues to characterize learning from the full history; Proposition 2(1) remains for separate derivation.